## Interchanging Rows Of Matrix Changes Sign Of Determinants!

Linear spaces and subspaces, linear transformations, bases, etc.
RedPrince007
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### Interchanging Rows Of Matrix Changes Sign Of Determinants!

Hello Everyone..!!
Now A Days I Am Learning About Matrix And Determinants And I Confused On One Properties Of Determinants Which Is: Interchanging Two Rows/Columns Of A Determinant Changes The Sign Of The Determinant!

My Question Is What Is The Logic(Reason) That -ve Sign Is Places Outside The Determinants While Interchanging Rows/Columns But No Sign Is Places Outsides In Gaussian Elimination (OR More Specific In Matrix)

I Don't Understand The Logic Behind This! I Google It A Lot But Found No Answer Under The Scope Of My Knowledge.! Can Anybody Please Explain Why We Do This..!! Thanks In Advance For Answer! And Sorry In Advance If I Am Not Specific..!!

buddy
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### Re: Interchanging Rows Of Matrix Changes Sign Of Determinant

Try doing a 2x2 and see what happens.

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```original: | -3 2 | | 1 5 | = (-3)(5) - (1)(2) = -15 - 2 = -17 swapped: | 2 -3 | | 5 1 | = (2)(1) - (5)(-3) = 2 - (-15) = 17```
Since any determinant can be done with 2x2s using minors and cofactors this shows how the sign changes: all the 2x2s flipped signs, so the whole thing does too. You probably could prove it with induction.

RedPrince007
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### Re: Interchanging Rows Of Matrix Changes Sign Of Determinant

Why We Place -ive Sign Outside Determinant It Is Clear To Me Now But My Second Question Is That We Place -ive Sign When We Interchange Rows Of Determinant But Why We Do Not Place -ive Sign When We Interchange Rows In Gaussian Elimination i.e., While Performing Row Operations!!

buddy
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### Re: Interchanging Rows Of Matrix Changes Sign Of Determinant

Its a different process for row ops on a matrix. Theres not "-1 to a power" for the matrix, only for the determinant. If you do something different & use a different process, you get different answers. Determinants aren't matrixes.